How many terms of the AP: -15, -13, -11, ldot | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $n$ be the number of terms required to make the sum $-55$.Here,the first term $(a) = -15$ and the common difference $(d) = -13 - (-15) = 2$.Th

Vedclass · Accepted Answer

$5, 11$

Vedclass · Answer

(A) Let $n$ be the number of terms required to make the sum $-55$.Here,the first term $(a) = -15$ and the common difference $(d) = -13 - (-15) = 2$.The sum of $n$ terms of an $AP$ is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.Substituting the values: $-55 = \frac{n}{2}[2(-15) + (n-1)2]$.$-55 = \frac{n}{2}[-30 + 2n - 2] = \frac{n}{2}[2n - 32] = n(n - 16)$.$n^2 - 16n + 55 = 0$.Factoring the quadratic equation: $n^2 - 11n - 5n + 55 = 0 \Rightarrow n(n - 11) - 5(n - 11) = 0$.$(n - 5)(n - 11) = 0$,so $n = 5$ or $n = 11$.The reason for the double answer is that the sum of the terms from the $6^{th}$ term to the $11^{th}$ term is zero. Specifically,the terms are $-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5$. The sum of the $6^{th}$ to $11^{th}$ terms is $(-5) + (-3) + (-1) + 1 + 3 + 5 = 0$. Thus,adding these terms to the sum of the first $5$ terms does not change the total sum.

Column $A$	Column $B$
$(A_{1}) \quad 2, -2, -6, -10, \ldots$	$(B_{1}) \quad \frac{2}{3}$
$(A_{2}) \quad a = -18, n = 10, a_{n} = 0$	$(B_{2}) \quad -5$
$(A_{3}) \quad a = 0, a_{10} = 6$	$(B_{3}) \quad 4$
$(A_{4}) \quad a_{2} = 13, a_{4} = 3$	$(B_{4}) \quad -4$
	$(B_{5}) \quad 2$
	$(B_{6}) \quad \frac{1}{2}$
	$(B_{7}) \quad 5$

How many terms of the $AP: -15, -13, -11, \ldots$ are needed to make the sum $-55$? Explain the reason for the double answer.

Similar Questions

In each of the following,$a$ and $d$ for an $A.P.$ are given. Find the $A.P.$ in each case. $a = \frac{15}{2}, \quad d = \frac{3}{2}$

If the $n$th terms of the two $APs$: $9, 7, 5, \ldots$ and $24, 21, 18, \ldots$ are the same,find the value of $n$. Also,find that term.

The sum of three numbers in $A.P.$ is $48$ and the sum of their squares is $800$. Find those numbers.

$Rs. 280$ is to be distributed among four prizes. If the prize amount in each successive prize reduces by $Rs. 20$,find the amounts paid in those prizes.

Vedclass Test Series

Exam Paper Generator

Online Exam Module